Diophantine Condition Research Articles

Lower-Dimensional Invariant Tori for Perturbations of a Class of Non-convex Hamiltonian Functions

We consider a class of quasi-integrable Hamiltonian systems obtained by adding to a non-convex Hamiltonian function of an integrable system a perturbation depending only on the angle variables. We focus on a resonant maximal torus of the unperturbed system, foliated into a family of lower-dimensional tori of codimension 1, invariant under a quasi-periodic flow with rotation vector satisfying some mild Diophantine condition. We show that at least one lower-dimensional torus with that rotation vector always exists also for the perturbed system. The proof is based on multiscale analysis and resummation procedures of divergent series. A crucial role is played by suitable symmetries and cancellations, ultimately due to the Hamiltonian structure of the system.

A solvable many-body problem, its equilibria, and a second-order ordinary differential equation whose general solution is polynomial

Some properties of a solvable N-body problem featuring several free parameters (“coupling constants”) are investigated. Restrictions on its parameters are reported which guarantee that all its solutions are completely periodic with a fixed period independent of the initial data (isochrony). The restrictions on its parameters which guarantee the existence of equilibria are also identified. In this connection a remarkable second-order ODE—generally not of hypergeometric type, hence not reducible to those characterizing the classical polynomials—is studied: if its parameters satisfy a Diophantine condition, its general solution is a polynomial of degree N, the N zeros of which identify the equilibria of the N-body system.

Open circle maps: small hole asymptotics

We consider escape from chaotic maps through a subset of phase space, the hole. Escape rates are known to be locally constant functions of the hole position and size. In spite of this, for the doubling map we can extend the current best result for small holes, a linear dependence on hole size h, to include a smooth h2 ln h term and explicit fractal terms to h2 and higher orders, confirmed by numerical simulations. For more general hole locations the asymptotic form depends on a dynamical Diophantine condition using periodic orbits ordered by stability.

On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems

In this paper we consider two-dimensional nonlinear quasi-periodic system with small perturbations. Assume that the unperturbed system has a hyperbolic-type degenerate equilibrium point and the frequency satisfies the Diophantine conditions. Using the KAM iteration we prove that for sufficiently small perturbations, the system can be reduced by a nonlinear quasi-periodic transformation to a suitable normal form with an equilibrium point at the origin. Hence, for the system we can obtain a small quasi-periodic solution.

Simultaneous linearization for commuting quasiperiodically forced circle diffeomorphisms

For commuting smooth quasiperiodically forced circle diffeomorphisms, we show that if the base frequencies and the fibred rotation numbers jointly satisfy some simultaneous Diophantine condition and if the diffeomorphisms are in some C ∞ C^{\infty } neighborhood of the corresponding rotations, then they are simultaneously C ∞ C^{\infty } -linearizable.

INTERACTION BETWEEN PERIODIC ELASTIC WAVES AND TWO CONTACT NONLINEARITIES

Propagation of elastic waves is studied in a 1D medium containing two cracks. The latter are modeled by smooth nonlinear jump conditions accounting for the finite, non-null compressibility of real cracks. The evolution equations are written in the form of a system of two nonlinear neutral delay differential equations, leading to a well-posed Cauchy problem. Perturbation analysis indicates that, under periodic excitation, the periodic solutions oscillate around positive mean values, which increase with the forcing level. This typically nonlinear phenomenon offers non-destructive means to evaluate the cracks. Existence, uniqueness and attractivity of periodic solutions is then examined. At some particular values of the ratio between the wave travel time and the period of the source, results are obtained whatever the forcing level. With a much larger set of ratios but at small forcing levels, results are obtained under a Diophantine condition. Lastly, numerical experiments are proposed to illustrate the behavior of the periodic diffracted waves.

Normalizations with Exponentially Small Remainders for Nonautonomous Analytic Periodic Vector Fields

In this paper we deal with analytic nonautonomous vector fields with a periodic time-dependancy, that we study near an equilibrium point. In a first part, we assume that the linearized system is split in two invariant subspaces E0 and E1. Under light diophantine conditions on the eigenvalues of the linear part, we prove that there is a polynomial change of coordinates in E1 allowing to eliminate up to a finite polynomial order all terms depending only on the coordinate u0 of E0 in the E1 component of the vector field. We moreover show that, optimizing the choice of the degree of the polynomial change of coordinates, we get an exponentially small remainder. In the second part, we prove a normal form theorem with exponentially small remainder. Similar theorems have been proved before in the autonomous case : this paper generalizes those results to the nonautonomous periodic case.

The Euclid Algorithm is totally gaussian

We consider Euclid’s gcd algorithm for two integers $(p, q)$ with $1 \leq p \leq q \leq N$, with the uniform distribution on input pairs. We study the distribution of the total cost of execution of the algorithm for an additive cost function $d$ on the set of possible digits, asymptotically for $N \to \infty$. For any additive cost of moderate growth $d$, Baladi and Vallée obtained a central limit theorem, and –in the case when the cost $d$ is lattice– a local limit theorem. In both cases, the optimal speed was attained. When the cost is non lattice, the problem was later considered by Baladi and Hachemi, who obtained a local limit theorem under an intertwined diophantine condition which involves the cost $d$ together with the “canonical” cost $c$ of the underlying dynamical system. The speed depends on the irrationality exponent that intervenes in the diophantine condition. We show here how to replace this diophantine condition by another diophantine condition, much more natural, which already intervenes in simpler problems of the same vein, and only involves the cost $d$. This “replacement” is made possible by using the additivity of cost $d$, together with a strong property satisfied by the Euclidean Dynamical System, which states that the cost $c$ is both “strongly” non additive and diophantine in a precise sense. We thus obtain a local limit theorem, whose speed is related to the irrationality exponent which intervenes in the new diophantine condition. We mainly use the previous proof of Baladi and Hachemi, and “just” explain how their diophantine condition may be replaced by our condition. Our result also provides a precise comparison between the rational trajectories of the Euclid dynamical system and the real trajectories.

Heaviness in toral rotations

We investigate the dimension of the set of points H(A, α) in the d-torus which have the property that their orbit under rotation by some α hits a fixed closed target A more often than expected for all finite initial portions of time. An upper bound for the lower Minkowski dimension of this set is found in terms of the upper Minkowski dimension of ∂A: $$\underline {\dim } _M (H(A,\alpha )) \leqslant \frac{{d + \overline {\dim } _M \partial {\rm A}}} {2},$$ which may be improved under certain Diophantine conditions on µ(A). In particular, in the event that µ(A) ∈ ℚ we have $$\underline {\dim } _M (H(A,\alpha )) \leqslant \overline {\dim } _M \partial {\rm A}.$$ The proof extends to translations in compact abelian groups more generally than just the torus, most notably the p-adic integers.

Asymptotic Property of Divergent Formal Solutions in Linearization of Singular Vector Fields

We study asymptotic properties of divergent formal solutions appearing in the linearization problem of a singular vector field without a Diophantine condition or existence of additional first integrals. We give an asymptotic meaning to divergent formal solutions constructed from a singular perturbative solution (cf. [6]).

Reducibility Results for Quasiperiodic Cocycles With Liouvillean Frequency

We study the reducibility problems for quasiperiodic cocycles in linear Lie groups with one frequency, irrespective of any Diophantine condition on the base dynamics. Under a non-degeneracy condition, a positive measure diagonalizable result is obtained for quasiperiodic \({GL(d,\mathbb R)}\) cocycles which are close to constants. It generalizes previous works by Avila–Fayad–Krikorian and Hou–You, and our approach is based on periodic approximation and KAM schemes.

Boundedness for the general semilinear Duffing equations via the twist theorem

In this paper, we prove the boundedness of all solutions for the periodic equation x ″ + ω 2 x + ϕ ( x ) = G x ( x , t ) + p ( t ) , where ω satisfies the Diophantine condition, ϕ ( x ) ∈ C 19 is bounded, p ( t ) ∈ C 23 and D x i D t j G ( x , t ) is bounded for 0 ⩽ i + j ⩽ 21 .

Global analytic, Gevrey and C∞ hypoellipticity on the 3‐torus

AbstractIt is proved that a class of sub‐Laplacians on the 3‐dimensional torus is globally analytic, Gevrey and C∞ hypoelliptic if and only if either a Diophantine condition holds or there is a point of finite type for the vector fields defining the operator under consideration. This work is motivated by the desire of attaining a better understanding of necessary and sufficient conditions for the global Gevrey and C∞ hypoellipticity of a sub‐Laplacian on a compact analytic manifold. This together with the more delicate local hypoellipticity are well‐known open problems in the theory of linear partial differential equations.

A Kam Scheme for SL(2, $${{\mathbb R}}$$ ) Cocycles with Liouvillean Frequencies

We develop a new KAM scheme that applies to SL(2, \({{\mathbb R}}\)) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg–Sinai’s theorem to arbitrary frequencies: under a closeness to constant assumption, the non-Abelian part of the classical reducibility problem can always be solved for a positive measure set of parameters.

Simultaneous linearization of germs of commuting holomorphic diffeomorphisms

AbstractLet α1,…,αn be irrational numbers which are linearly independent over the rationals, and f1,…,fn commuting germs of holomorphic diffeomorphisms in ℂ such that fk(0)=0,f′k(0)=e2πiαk,k=1,…,n. Moser showed that f 1,…,fn are simultaneously linearizable (i.e. conjugate by a germ of holomorphic diffeomorphism h to the rigid rotations Rαk(z)=e2πiαkz) if the vector of rotation numbers (α 1,…,αn) satisfies a Diophantine condition. Adapting Yoccoz’s renormalization to the setting of commuting germs, we show that simultaneous linearization holds in the presence of a weaker Brjuno-type condition ℬ(α1,…,αn)<+∞, where ℬ(α1,…,αn) is a multivariable analogue of the Brjuno function. If there are no periodic orbits for the action of the germs f1,…,fn in a neighbourhood of the origin, then a weaker arithmetic condition ℬ′ (α 1,…,αn)<+∞ analogous to Perez-Marco’s condition for linearization in the absence of periodic orbits is shown to suffice for linearizability. Normalizing the germs to be univalent on the unit disc, in both cases the Siegel discs are shown to contain discs of radii $C e^{-2\pi \mathcal {B}}, C e^{-2\pi \mathcal {B'}}$ for some universal constant C.

Sectorial solution to semilinear singular partial differential equation

We shall show the solvability of semilinear Fuchsian partial differential systems in a multi-sectorial domain. Our equation contains a linearizing equation of a singular vector field to its linear part when so-called small denominators occur. We will show the existence of an analytic solution in a multi-sectorial domain without assuming any Diophantine condition.

Exact and explicit solutions for a nonlinear extended iterative differential equation

This paper is concerned with a nonlinear iterative functional differential equation x ′( z ) = 1/ x ( p ( z ) + bx ′( z )). By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. We discuss not only in the general case, but also in critical cases, especially for α given in Schröder transformation is a root of the unity. And in case (H4), we dealt with the equation under the Brjuno condition, which is weaker than the Diophantine condition. Moreover, the exact and explicit solution of the original equation has been investigated for the first time. Such equations are important in both applications and the theory of iterations.

Exponential Riesz Bases, Discrepancy of Irrational Rotations and BMO

We study the basis property of systems of exponentials with frequencies belonging to ‘simple quasicrystals’. We show that a diophantine condition is necessary and sufficient for such a system to be a Riesz basis in L 2 on a finite union of intervals. For the proof we extend to BMO a theorem of Kesten about the discrepancy of irrational rotations of the circle.

Analytic Solutions of a Nonhomogeneous Iterative Functional Differential Equation

This paper is concerned with an iterative functional differential equation with state-dependent delay As well as in previous works, we reduce this problem with the Schroeder transformation to obtain auxiliary equation. For technical reasons, in previous work the constantgiven in the Schroeder transformation, is required to fulfill that is off the unit circle or lies on the circle with the Diophantine condition. In this paper, we discuss not only thoseat a root of the unity, but also those near resonance under the Brjuno condition.

Analytic Solutions of an Iterative Functional Differential Equation

This paper is concerned with the existence of analytic solutions of an iterative functional differential equation. By constructing a convergent power series solution of an auxiliary equation, analytic solutions for the original differential equation are obtained. We discuss not only those α at the hyperbolic case 0<|α|<1 but also those α near resonance under Diophantine condition.